Document Type : Research Paper
Authors
1 Department of Statistics, Faculty of Mathematical Sciences and Statis- tics, University of Birjand, Birjand, Iran
2 Department of Engineering Science, College of Engineering, Univer- sity of Tehran, Tehran, P.O. Box 11365-4563, Iran
Abstract
A new approach to define the similarity measure between
interval-valued fuzzy sets is presented. The proposed approach is
based on a weighted measure in which the normalized similarities
between lower functions and also between upper functions are
combined by a weight parameter. The properties of this similarity
measure are investigated. It is shown that, the proposed measure
has some advantages in comparison with the commonly used
similarity measures.
Keywords
\bibitem{AT1} K. Atanassov, {\it Intuitionistic fuzzy sets}, Fuzzy
Sets and Systems, {\bf20} (1986), 87-96.
\bibitem{AT} K. Atanassov, Intuitionistic Fuzzy Sets: Theory and
Applications, Springer, Heidelberg, 1999.
\bibitem{AT2} K. Atanassov, {\it New operations defined over the intuitionistic
fuzzy sets}, Fuzzy Sets and Systems, {\bf61} (1994), 137-142.
\bibitem{ATgar} K. Atanassov and G. Gargov, {\it Interval valued intuitionistic
fuzzy sets}, Fuzzy Sets and Systems, {\bf31} (1989), 343-349.
\bibitem{bb} H. Bustince and P. Burillo, {\it Vague sets are intuitionistic fuzzy sets},
Fuzzy Sets and Systems, {\bf79} (1996), 403-405.
\bibitem{chata} J. Chachi and S. M. Taheri, {\it A unified approach to similarity measures
between intuitionistic fuzzy sets}, International Journal of
Intelligent Systems, {\bf28} (2013), 669-685.
\bibitem{chen} S. M. Chen, {\it Measures of similarity between vague
sets}, Fuzzy Sets and Systems, {\bf74} (1995), 217-223.
\bibitem{chen121} S. M. Chen, {\it Similarity measure between vague sets and
between elements.}, IEEE Trans. Systems Man Cybernt., {\bf27}
(1997), 153-158.
\bibitem{chenli} Y. Chen and B. Li, {\it Dynamic multi-attribute decision making model based on triangular intuitionistic fuzzy
numbers}, Scientia Iranica, {\bf18} (2011), 268-274.
\bibitem{chentan} S. M. Chen and J. M. Tan, {\it Handling multicriteria fuzzy decision-making problems
based on vague set theory}, Fuzzy Sets and Systems, {\bf67}
(1994), 163-172.
\bibitem{de100} S. K. De, R. Biswas and A. R. Roy, {\it An application of intuitionistic fuzzy
sets in medical diagnosis}, Fuzzy Sets and Systems, {\bf117}
(2001), 209-213.
\bibitem{dian} D. S. Dinagar and A. Anbalagan, {\it A new similarity measure between type-2 fuzzy numbers and fuzzy risk
analysis}, Iranian Journal of Fuzzy Systems, {\bf10} (2013),
79-95.
%\bibitem{dgech} A. Dziech and M. B. Gorzalczany, {\it Decision making in signal transmission problems with
%interval-valued fuzzy sets}, Fuzzy Sets and Systems, {\bf23}
%(1987), 191-203.
\bibitem{Farhad} B. Farhadinia, {\it An
efficient similarity measure for intuitionistic fuzzy sets}, Soft
Computing, {\bf18} (2014), 85-94.
\bibitem{gaubu} W. L. Gau and D. J. Buehrer, {\it Vague sets},
IEEE Transactions on Systems, Man, and Cybernetics, {\bf23}
(1993), 610-614.
\bibitem{gor} M. B. Gorzalczany, {\it An Interval-valued fuzzy inference method - some basic
properties}, Fuzzy Sets and Systems, {\bf31} (1989), 243-251.
\bibitem{gorz} M. B. Gorzalczany, {\it Approximate inference with interval-valued fuzzy sets - an outline},
In: Proceeding Polish. Symp. on Interval and Fuzzy Mathematics,
Poznan, (1983), 89-95.
\bibitem{hipel} K. W. Hipel, D. M. Kilgour and M. Abul
Bashar, {\it Fuzzy preferences in multiple participant decision
making}, Scientia Iranica, {\bf18} (2011), 627-638.
\bibitem{hunya} W. L. Hung and M. Sh. Yang, {\it Similarity measures of intuitionistic fuzzy sets
based on Hausdorff distance}, Pattern Recognition Letters,
{\bf25} (2004), 1603-1611.
\bibitem{hunya2} W. L. Hung and M. Sh. Yang, {\it Similarity measures of intuitionistic fuzzy sets
based on $L_p$ metric}, International Journal of Approximate
Reasoning, {\bf46} (2007), 120-136.
\bibitem{htal10} Ch. M. Hwang, M. Sh. Yang, W. L. Hung and M. G. Lee,
{\it A similarity measure of intuitionistic fuzzy sets based on
the Sugeno integral with its application to pattern recognition},
Information Sciences, {\bf189} (2012), 93-109.
\bibitem{juletal} P. Julian, K. Ch. Hung and Sh. J. Lin, {\it On the Mitchell similarity measure and its application to pattern
recognition,} Pattern Recognition Letters, {\bf33} (2012),
1219-1223.
\bibitem{lich} D. Li and C. Cheng, {\it New similarity measures of intuitionistic fuzzy
sets and application to pattern recognition}, Pattern Recognition
Letters, {\bf23} (2002), 221-225.
\bibitem{lishi} Z. Liang and P. Shi, {\it Similarity measures on intuitionistic fuzzy
sets}, Pattern Recognition Letters, {\bf24} (2003), 2687-2693.
\bibitem{mitch} H. B. Mitchell, {\it On the Dengfeng-Chuitian similarity
measure and its application to pattern recognition}, Pattern
Recognition Letters, {\bf24} (2003), 3101-3104.
\bibitem{rabetal} M. R. Rabiei, N. R. Arghami, S. M.
Taheri and B. S. Gildeh, {\it Least-squares approach to
regression modeling in full interval-valued fuzzy environment},
Soft Computing, {\bf 18} (2014), 2043-2059.
\bibitem{robi} M. K. Roy and R. Biswas, {\it I-V fuzzy relations and Sanchez's approach for
medical diagnosis}, Fuzzy Sets and Systems, {\bf47} (1992), 35-38.
%\bibitem{Szmidt} E. Szmidt and J. Kacprzyk, {\it Intuitionistic fuzzy sets for more realistic group decision making},
%Proceeding of TRANSITION'97, 18-21 June, Warsaw, Poland, (1997),
%430-433.
\bibitem{turk} B. Turksen, {\it Interval valued fuzzy sets based on normal forms},
Fuzzy Sets and Systems, {\bf20} (1986), 191-210.
\bibitem{waxin} W. Wang and X. Xin, {\it Distance measure between intuitionistic
fuzzy sets}, Pattern Recognition Letters, {\bf26} (2005),
2063-2069.
\bibitem{weietal} C. P. Wei, P. Wang and Y. Zh. Zhang,
{\it Entropy, similarity measure of interval-valued
intuitionistic fuzzy sets and their applications}, Information
Sciences, {\bf181} (2011), 4273-4286.
\bibitem{weim} G. W. Wei and J. M. Merig${\rm\acute{o}}$, {\it Methods for strategic decision-making problems
with immediate probabilities in intuitionistic fuzzy setting},
Scientia Iranica, {\bf19} (2012), 1936-1946.
\bibitem{xu} Z. Xu, {\it A method based on distance measure for interval-valued
intuitionistic fuzzy group decision making}, Information Sciences,
{\bf180} (2010), 181-190.
\bibitem{ye234} J. Ye, {\it Cosine similarity measures for intuitionistic fuzzy
sets and their applications}, Mathematical and Computer
Modelling, {\bf53} (2011), 91-97.
\bibitem{zadeh} L. A. Zadeh, {\it Fuzzy sets}, Information and Control, {\bf8} (1965), 338-356.
\bibitem{zeng} W. Zeng and P. Guo, {\it Normalized distance, similarity measure, inclusion
measure and entropy of interval-valued fuzzy sets and their
relationship}, Information Sciences, {\bf178} (2008), 1334-1342.
\bibitem{zeli} W. Zeng and H. Li, {\it Relationship between similarity measure and entropy of interval
valued fuzzy sets}, Fuzzy Sets and Systems, {\bf157} (2006),
1477-1484.
\bibitem{zhletal} Q. Zhang, F. Liu, L. Wu and Sh. Luo,
{\it Information entropy, similarity measure and inclusion
measure of intuitionistic fuzzy sets}, Communications in Computer
and Information Science, {\bf307} (2012), 392-398.
Sets and Systems, {\bf20} (1986), 87-96.
\bibitem{AT} K. Atanassov, Intuitionistic Fuzzy Sets: Theory and
Applications, Springer, Heidelberg, 1999.
\bibitem{AT2} K. Atanassov, {\it New operations defined over the intuitionistic
fuzzy sets}, Fuzzy Sets and Systems, {\bf61} (1994), 137-142.
\bibitem{ATgar} K. Atanassov and G. Gargov, {\it Interval valued intuitionistic
fuzzy sets}, Fuzzy Sets and Systems, {\bf31} (1989), 343-349.
\bibitem{bb} H. Bustince and P. Burillo, {\it Vague sets are intuitionistic fuzzy sets},
Fuzzy Sets and Systems, {\bf79} (1996), 403-405.
\bibitem{chata} J. Chachi and S. M. Taheri, {\it A unified approach to similarity measures
between intuitionistic fuzzy sets}, International Journal of
Intelligent Systems, {\bf28} (2013), 669-685.
\bibitem{chen} S. M. Chen, {\it Measures of similarity between vague
sets}, Fuzzy Sets and Systems, {\bf74} (1995), 217-223.
\bibitem{chen121} S. M. Chen, {\it Similarity measure between vague sets and
between elements.}, IEEE Trans. Systems Man Cybernt., {\bf27}
(1997), 153-158.
\bibitem{chenli} Y. Chen and B. Li, {\it Dynamic multi-attribute decision making model based on triangular intuitionistic fuzzy
numbers}, Scientia Iranica, {\bf18} (2011), 268-274.
\bibitem{chentan} S. M. Chen and J. M. Tan, {\it Handling multicriteria fuzzy decision-making problems
based on vague set theory}, Fuzzy Sets and Systems, {\bf67}
(1994), 163-172.
\bibitem{de100} S. K. De, R. Biswas and A. R. Roy, {\it An application of intuitionistic fuzzy
sets in medical diagnosis}, Fuzzy Sets and Systems, {\bf117}
(2001), 209-213.
\bibitem{dian} D. S. Dinagar and A. Anbalagan, {\it A new similarity measure between type-2 fuzzy numbers and fuzzy risk
analysis}, Iranian Journal of Fuzzy Systems, {\bf10} (2013),
79-95.
%\bibitem{dgech} A. Dziech and M. B. Gorzalczany, {\it Decision making in signal transmission problems with
%interval-valued fuzzy sets}, Fuzzy Sets and Systems, {\bf23}
%(1987), 191-203.
\bibitem{Farhad} B. Farhadinia, {\it An
efficient similarity measure for intuitionistic fuzzy sets}, Soft
Computing, {\bf18} (2014), 85-94.
\bibitem{gaubu} W. L. Gau and D. J. Buehrer, {\it Vague sets},
IEEE Transactions on Systems, Man, and Cybernetics, {\bf23}
(1993), 610-614.
\bibitem{gor} M. B. Gorzalczany, {\it An Interval-valued fuzzy inference method - some basic
properties}, Fuzzy Sets and Systems, {\bf31} (1989), 243-251.
\bibitem{gorz} M. B. Gorzalczany, {\it Approximate inference with interval-valued fuzzy sets - an outline},
In: Proceeding Polish. Symp. on Interval and Fuzzy Mathematics,
Poznan, (1983), 89-95.
\bibitem{hipel} K. W. Hipel, D. M. Kilgour and M. Abul
Bashar, {\it Fuzzy preferences in multiple participant decision
making}, Scientia Iranica, {\bf18} (2011), 627-638.
\bibitem{hunya} W. L. Hung and M. Sh. Yang, {\it Similarity measures of intuitionistic fuzzy sets
based on Hausdorff distance}, Pattern Recognition Letters,
{\bf25} (2004), 1603-1611.
\bibitem{hunya2} W. L. Hung and M. Sh. Yang, {\it Similarity measures of intuitionistic fuzzy sets
based on $L_p$ metric}, International Journal of Approximate
Reasoning, {\bf46} (2007), 120-136.
\bibitem{htal10} Ch. M. Hwang, M. Sh. Yang, W. L. Hung and M. G. Lee,
{\it A similarity measure of intuitionistic fuzzy sets based on
the Sugeno integral with its application to pattern recognition},
Information Sciences, {\bf189} (2012), 93-109.
\bibitem{juletal} P. Julian, K. Ch. Hung and Sh. J. Lin, {\it On the Mitchell similarity measure and its application to pattern
recognition,} Pattern Recognition Letters, {\bf33} (2012),
1219-1223.
\bibitem{lich} D. Li and C. Cheng, {\it New similarity measures of intuitionistic fuzzy
sets and application to pattern recognition}, Pattern Recognition
Letters, {\bf23} (2002), 221-225.
\bibitem{lishi} Z. Liang and P. Shi, {\it Similarity measures on intuitionistic fuzzy
sets}, Pattern Recognition Letters, {\bf24} (2003), 2687-2693.
\bibitem{mitch} H. B. Mitchell, {\it On the Dengfeng-Chuitian similarity
measure and its application to pattern recognition}, Pattern
Recognition Letters, {\bf24} (2003), 3101-3104.
\bibitem{rabetal} M. R. Rabiei, N. R. Arghami, S. M.
Taheri and B. S. Gildeh, {\it Least-squares approach to
regression modeling in full interval-valued fuzzy environment},
Soft Computing, {\bf 18} (2014), 2043-2059.
\bibitem{robi} M. K. Roy and R. Biswas, {\it I-V fuzzy relations and Sanchez's approach for
medical diagnosis}, Fuzzy Sets and Systems, {\bf47} (1992), 35-38.
%\bibitem{Szmidt} E. Szmidt and J. Kacprzyk, {\it Intuitionistic fuzzy sets for more realistic group decision making},
%Proceeding of TRANSITION'97, 18-21 June, Warsaw, Poland, (1997),
%430-433.
\bibitem{turk} B. Turksen, {\it Interval valued fuzzy sets based on normal forms},
Fuzzy Sets and Systems, {\bf20} (1986), 191-210.
\bibitem{waxin} W. Wang and X. Xin, {\it Distance measure between intuitionistic
fuzzy sets}, Pattern Recognition Letters, {\bf26} (2005),
2063-2069.
\bibitem{weietal} C. P. Wei, P. Wang and Y. Zh. Zhang,
{\it Entropy, similarity measure of interval-valued
intuitionistic fuzzy sets and their applications}, Information
Sciences, {\bf181} (2011), 4273-4286.
\bibitem{weim} G. W. Wei and J. M. Merig${\rm\acute{o}}$, {\it Methods for strategic decision-making problems
with immediate probabilities in intuitionistic fuzzy setting},
Scientia Iranica, {\bf19} (2012), 1936-1946.
\bibitem{xu} Z. Xu, {\it A method based on distance measure for interval-valued
intuitionistic fuzzy group decision making}, Information Sciences,
{\bf180} (2010), 181-190.
\bibitem{ye234} J. Ye, {\it Cosine similarity measures for intuitionistic fuzzy
sets and their applications}, Mathematical and Computer
Modelling, {\bf53} (2011), 91-97.
\bibitem{zadeh} L. A. Zadeh, {\it Fuzzy sets}, Information and Control, {\bf8} (1965), 338-356.
\bibitem{zeng} W. Zeng and P. Guo, {\it Normalized distance, similarity measure, inclusion
measure and entropy of interval-valued fuzzy sets and their
relationship}, Information Sciences, {\bf178} (2008), 1334-1342.
\bibitem{zeli} W. Zeng and H. Li, {\it Relationship between similarity measure and entropy of interval
valued fuzzy sets}, Fuzzy Sets and Systems, {\bf157} (2006),
1477-1484.
\bibitem{zhletal} Q. Zhang, F. Liu, L. Wu and Sh. Luo,
{\it Information entropy, similarity measure and inclusion
measure of intuitionistic fuzzy sets}, Communications in Computer
and Information Science, {\bf307} (2012), 392-398.
)